Subgroups — #94576

Order 512.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13628	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₆₄	C₂	1	2	yes	maximal
3	C₂×M₆₄	C₄	1	1	yes	characteristic
4	C₂²×C₃₂	C₂²	1	1	yes	characteristic
5	#2108	C₄	1	1	yes	characteristic
6	C₂×C₆₄	—	2	2	no	—
7	C₄⋊C₃₂	—	2	2	no	—
8	C₂×M₃₂	C₈	1	1	yes	characteristic
9	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
10	#405	C₈	1	1	yes	characteristic
11	M₆₄	—	4	1	no	—
12	C₂×C₃₂	—	2	1	no	—
13	C₂×C₃₂	—	4	1	no	—
14	C₂×C₃₂	D₄	1	2	yes	—
15	#346	—	2	1	no	—
16	C₄×C₁₆	—	2	1	no	—
17	C₆₄	—	4	2	no	—
18	C₂×M₁₆	C₁₆	1	1	yes	characteristic
19	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
20	#117	C₁₆	1	1	yes	characteristic
21	M₃₂	—	4	1	no	—
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₁₆	M₁₆	1	1	yes	characteristic
27	#97	—	2	1	no	—
28	C₄×C₈	—	2	1	no	—
29	C₃₂	—	2	2	no	—
30	C₃₂	—	4	1	no	—
31	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
32	M₁₆	—	4	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₈	M₃₂	1	1	yes	characteristic
38	C₄⋊C₄	C₃₂	1	2	yes	—
39	C₂²⋊C₄	—	2	1	no	—
40	C₄²	—	2	1	no	—
41	C₁₆	—	2	1	no	—
42	C₁₆	—	4	1	no	—
43	C₁₆	#99	1	1	yes	characteristic
44	C₁₆	#101	1	1	yes	characteristic
45	C₂³	M₆₄	1	1	yes	characteristic
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
50	C₂×C₄	C₂×C₃₂	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	—	4	1	no	—
53	C₈	#324	1	1	yes	characteristic
54	C₈	#349	1	1	yes	characteristic
55	C₂²	—	2	2	no	—
56	C₂²	—	2	1	no	—
57	C₂²	C₂²⋊C₃₂	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	4	2	no	—
60	C₄	#1322	1	1	yes	characteristic
61	C₄	#1400	1	1	yes	characteristic
62	C₂	—	4	1	no	—
63	C₂	#7362	1	1	yes	characteristic
64	C₂	#7514	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃₂
Frattini, cosocle	C₂²×C₃₂	C₂²
socle	C₂²	C₂²⋊C₃₂
prime radical	C₂³	M₆₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13628	2
C₂²⋊C₆₄	2